Thermal Entanglement and Thermal Discord in two-qubit Heisenberg XYZ Chain with Dzyaloshinskii-Moriya Interactions

In order to explore the effect of external temperature $T$ in quantum correlation we compute thermal entanglement and thermal discord analytically in the Heisenberg $X$ $Y$ $Z$ model with Dzyaloshinskii-Moriya Interaction term ${\bm D} \cdot \left( {\bm \sigma}_1 \times {\bm \sigma}_2 \right)$. For the case of thermal entanglement it is shown that quantum phase transition occurs at $T = T_c$ due to sudden death phenomenon. For antiferromagnetic case the critical temperature $T_c$ increases with increasing $|{\bm D}|$. For ferromagnetic case, however, $T_c$ exhibits different behavior in the regions $|{\bm D}| \geq |{\bm D_*}|$ and $|{\bm D}| < |{\bm D_*}|$, where ${\bm D_*}$ is particular value of ${\bm D}$. It is shown that $T_c$ becomes zero at $|{\bm D}| = |{\bm D_*}|$. We explore the behavior of thermal discord in detail at $T \approx T_c$. For antiferromagnetic case the external temperature makes the thermal discord exhibit exponential damping behavior, but it never reaches to exact zero. For ferromagnetic case the thermal entanglement and thermal discord are shown to be zero simultaneously at $T_c = 0$ and $|{\bm D}| = |{\bm D_*}|$. This is unique condition for simultaneous disappearance of thermal entanglement and thermal discord in this model.